# Probability and Statistics Seminar

## Past sessions

Newer session pages: Next 10 9 8 7 6 5 4 3 2 1 Newest

### Extremes of Volterra Series Expansions with Heavy-Tailed Innovations

Linear time series models have widely been used in many areas of science. However, it is becoming increasingly clear that linear models do not represent adequately features often exhibited by data sets from environmental and economical processes. One property which is important and needs exploration is the study of sudden bursts of large positive and negative values the sample paths of non-linear processes generally produce.

There is no unifying theory that is applicable to all non-linear systems and consequently the study of such systems has to be restricted to special classes of non-linear models. Volterra series expansions or polynomial systems are one such class. It is well known that Volterra series expansions are known to be the most general non-linear representation for any stationary sequence.

Volterra series expansions have found significant applications in signal processing such as image, video, and speech processing where the non-linearities are mild enough to allow approximations by low order polynomial approximations. In view of these facts, the study of the extremal properties of finite order Volterra series expansions would be highly valuable in better understanding the extremal properties of non-linear processes as well as understanding the order of identification and adequacy of Volterra series, when used as models in signal processing. In fact, such extremal properties may suggest a way of finding finite order Volterra expansions which is consistent with the non-linearities of the observed process.

In this work, we look at the extremal behavior of Volterra series expansions generated by heavy-tailed innovations, via a point process formulation. We also look at some examples of bilinear processes and compare the known results on extremes of such processes with the corresponding adequate Volterra series approximations. A way of determining the proper order of a finite Volterra expansion and hence a finite order polynomial model for the observed non-linear data is also suggested.

### Robust Estimation of the Parameters in the FANOVA Model

We present a robust approach for fitting models with both additive and multiplicative effects of two-way tables. The method is an extension of the median polish technique and uses a robust alternating regression algorithm. The approach is highly robust, and also works well when there are more variables than observations. The model can be simplified to a purely additive or multiplicative model, the latter allows to construct a robust biplot being not predetermined by outliers. The performance of the method will be illustrated by real and artificial examples.

### Uso de Métodos MCMC em Análise Bayesiana de Dados de Sobrevivência

A análise de dados de sobrevivência em geral apresenta dados incompletos e presença de covariáveis. O uso de modelos paramétricos para esses dados pode envolver um grande número de parâmetros. Além disso, os modelos paramétricos usuais podem não ser adequados para muitos conjuntos de dados, pois a função de risco pode ter formas diversas como forma de banheira, multimodalidade, riscos crescentes ou decrescentes, entre outras. Por isso, consideramos modelos mais complexos como modelos de misturas de distribuições paramétricas na presença ou não de covariáveis.

O uso de métodos bayesianos para esses modelos pode ser muito simplificado a partir de métodos de simulação de Monte Carlo em Cadeias de Markov (MCMC), como os algoritmos de Gibbs sampling e Metropolis-Hastings. Também podemos usar critérios bayesianos para discriminar diferentes modelos usados para os dados de sobrevivência, usando estimativas de Monte Carlo a partir das amostras geradas pelo algoritmo Gibbs sampling para a função preditiva. Vários exemplos com dados reais serão considerados para ilustrar a metodologia proposta.

### Taxas de Alarme em Esquemas de Controlo de Qualidade

O desempenho de esquemas de controlo de qualidade é usualmente avaliado à custa de características do run length (RL) - o número de amostras recolhidas até à emissão de um alarme. O average run length (ARL) é de longe a mais popular dessas características e tem sido - extensiva e incorrectamente - utilizado na literatura para descrever o desempenho de um esquema de controlo.

O uso da função taxa de falha de RL foi proposto por Margavio et al. (1995), e quando avaliada em $m=1,2,\dots$, representa a probabilidade de ser emitido alarme pela amostra $m$, sabendo que as $m-1$ amostras anteriores não foram responsáveis pela emissão desse alarme.

Esta função pode ser entendida como uma taxa de alarme, fornece um retrato condicional e mais completo do desempenho de esquemas e será estudada para alguns esquemas de controlo do tipo markoviano.

Daremos destaque à influência das matrizes estocasticamente monótonas no comportamento da taxa de alarme e ilustraremos alguns resultados numéricos e estocásticos que lhe dizem respeito. De notar que tais resultados estocásticos permitem avaliar, por exemplo, o impacto da adopção de head starts no desempenho de esquemas de controlo, de modo qualitativo e mais objectivo.

#### Referências

• Margavio, T.M., Conerly, M.D., Woodall, W.H. and Drake, L.G. (1995). Alarm rates for quality control charts. StatisticsProbability Letters 24, 219-224.

### Modelação Estocástica para Redes sem Fios da Próxima Geração

As redes sem fios da próxima geração são vistas hoje em dia como um dos factores chaves para o desenvolvimento da infra-estrutura de comunicação global emergente. O seu desenho, planeamento e controlo devem ser suportados por modelos de tráfego adequados, nos quais a mobilidade e os novos aspectos de teletráfego serão considerados de uma forma integrada.

### Multiplicative Survival Models based on Counting Processes

Modern survival analysis may be effectively handled within the mathematical framework of counting processes. This theory introduced by Aalen (1972) has been the subject of intense research ever since. In this setting, emphasis is given to construction of the likelihood function due to its importance for both frequentist and Bayesian analysis. Some multiplicative survival models are here presented from a Bayesian perspective, especially frailty models for univariate survival data. A gamma process with independent increments in disjoint intervals is used to model the prior process of the baseline hazard function, and the frailty distribution is assumed to be a gamma distribution. Markov Chain Monte Carlo methods are used to find estimates of several quantities of interest. At last, this approach is illustrated with one example.

### Simultaneous and Multivariate Control Charts for Time Series Data

In this talk we present several new control charts for univariate as well as multivariate time series. All control schemes are EWMA (exponentially weighted moving average) charts.

First, simultaneous control schemes for the mean and the autocovariances of a univariate stationary process are introduced. A multivariate quality characteristic is considered. This quantity is transformed to a one-dimensional variable by using the Mahalanobis distance. The test statistic is obtained by smoothing this variable. Another control chart is based on a multivariate EWMA attempt which is directly applied to our quality characteristic. After that the resulting statistic is transformed to a univariate random variable. Besides modified control charts we consider residual charts, too. In an extensive simulation study all control schemes are compared with each other. The target process is assumed to be an ARMA(1,1) process with normal white noise.

EWMA control charts for multivariate time series were discussed by Kramer and Schmid (1997). Their aim was to find deviations in the mean behaviour. Here we focus on charts detecting changes between the cross-covariances of a multivariate stationary process. The starting point is again a multivariate characteristic. To introduce control charts a similar procedure is chosen as described above. In our comparison study the target process is taken as a 4-variate AR(1) process.

#### References

• Kramer, H. and Schmid, W. (1997). EWMA charts for multivariate time series. Sequential Analysis 16, 131-154.
• Rosolowski, M. and Schmid, W. (2001). EWMA charts for monitoring the mean and the autocovariances of stationary Gaussian process. Submitted for publication.
• Schmid, W. and Sliwa, P. (2001). Monitoring the cross-covariances of a multivariate time series. Technical Report.

### Métodos robustos en Análisis Multivariado

Se compararán las virtudes y defectos de los distintos enfoques y se mostrarán algunas aplicaciones.

### Silence Diagnostics: The Influence Curve Revisited

The influence curve (Hampel, 1974), latterly known as the influence function, lies at the heart of an established approach to robust statistics. Finite sample versions of it also form the foundation of that part of diagnostics known as influence analysis. In their booklength exposition, Hampel at al. (1986), while providing a formal mathematical derivation, emphasise that:

“The importance of the influence function lies in its heuristic interpretation: it describes the effect of an infinitesimal contamination at the point $x$ on the estimate, standardised by the amount of contamination.”

The present talk explores the extent to which this heuristic interpretation can itself be formalised. It is based on developments of the concept of salience and of the perturbation geometry introduced in Critchely et al. (2001).

#### References

• Critchley, F., Atkinson, R.A., Lu, G. and Biazi, E. (2001). Influence analysis based on the case sensitivity function. J. Royal Statist. Soc., B. 63(2), 307-323.
• Hampel, F.R. (1974). The influence curve and its role in robust estimation. J. Am. Satist. Soc., 69, 383-393.
• Hampel, F.R., Ronchetti, E.M. Rousseeuw, P.J. and Stahel, W.A. (1986). Robust Statistics: The Approach Based on Influence Functions. Wiley.